Final answer:
The surface area of the triangular prism with an equilateral triangle base is approximately 279.23 square inches, calculated by finding the areas of the triangle faces and rectangular faces and summing them.
Step-by-step explanation:
To find the surface area of a triangular prism with an equilateral triangle base, we need to calculate the area of the two triangular faces and the three rectangular faces. The surface area (SA) can be found using the formula:
SA = (base area of triangle × 2) + (perimeter of triangle × height of prism)
For an equilateral triangle, all sides are equal, so the base area (A) of the triangle is:
A = ½ × base × height
Given the base (b) is 6.1 inches and using the Pythagorean theorem, the height (h) of the triangle can be found since it forms a 30°-60°-90° right triangle:
h = (sqrt(3)/2) × b
h = (sqrt(3)/2) × 6.1 in
≈ 5.28 in
Now, we can find the area of one triangular face:
A = ½ × 6.1 in × 5.28 in
≈ 16.09 in²
The perimeter (P) of the triangle is:
P = 3 × b
= 3 × 6.1 in
= 18.3 in
The height (H) of the prism is given as 13.5 inches. So, the area of the three rectangles is:
Area of rectangles = P × H
= 18.3 in × 13.5 in
= 247.05 in²
Finally, the total surface area of the prism is:
SA = (2 × 16.09 in²) + 247.05 in²
≈ 279.23 in²
The surface area of the triangular prism is approximately 279.23 square inches.